continuous_combiner_bilinear
plain-language theorem explainer
The continuous-combiner translation theorem shows that any comparison operator obeying the four Aristotelian conditions, scale invariance, and continuous route independence yields a bilinear combiner of the form P(u, v) = 2u + 2v + c u v once an auxiliary package of mollifier smoothness, second-derivative identity, and psi-affine completion is supplied. Researchers extending the Recognition Science functional equation beyond polynomial regularity would cite this to obtain the RCL under weaker hypotheses. The proof is a direct one-line wrapper,
Claim. Let $C$ be a comparison operator satisfying the continuous Law of Logic (identity, non-contradiction, excluded middle, scale invariance, continuous route independence, and non-triviality). Suppose $C$ also satisfies the continuous combiner analysis inputs (finite mollifier smoothness, second-derivative input from the log-smoothness bootstrap, and psi-affine completion). Then there exist $P : ℝ → ℝ → ℝ$ and $c ∈ ℝ$ such that for all $x, y > 0$, derivedCost$(C, xy) +$ derivedCost$(C, x/y) = P($derivedCost$(C, x),$ derivedCost$(C, y))$ and $P(u, v) = 2u + 2v + c u v$ for all real $u, v$.
background
The GeneralizedDAlembert module replaces the polynomial-degree-≤-2 hypothesis of the original Translation Theorem with continuity of the combiner. SatisfiesLawsOfLogicContinuous augments the standard SatisfiesLawsOfLogic by substituting ContinuousRouteIndependence for the polynomial version. ContinuousCombinerAnalysisInputs is an explicit hypothesis package containing finite smoothness via mollifiers, a second-derivative identity, and psi-affine completion; the module documentation states that the quartic-log obstruction shows this package is not automatic from continuity alone.
proof idea
The proof is a one-line wrapper that applies continuous_combiner_bilinear_classification to the supplied comparison operator $C$, the continuous Law of Logic witness, and the analysis inputs. This classification lemma assembles the mollifier smoothness and derivative conditions into the explicit bilinear expression for the combiner $P$.
why it matters
This result supplies the continuous-combiner form of the Translation Theorem and is invoked directly by the downstream theorem RCL_is_unique_functional_form_of_logic_continuous. It recovers the bilinear combiner that encodes the Recognition Composition Law $J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)$ under the continuous hypothesis package. The module documentation notes that the quartic-log counterexample leaves finite pairwise polynomial closure as the sharp hypothesis for the Law-of-Logic paper and touches the open question of whether weaker regularity can replace the current analysis package.
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